Markov Rewards Processes with Impulse Rewards and Absorbing States

We study the expected accumulated reward for a discrete-time Markov reward model with absorbing states. The rewards are impulse rewards, where a reward $\rho_{ij}$ is accumulated when transitioning from state $i$ to state $j$. We derive an explicit, single-letter expression for the expected accumulated reward as a function of the number of time steps $n$ and include in our analysis the limit in which $n \to \infty$.